Objective. Our goal in this paper is then to construct learning algorithms that maximize \({\rm AUC}^{\rm {ovo}}\) . To fit in the standard machine learning paradigm,
we follow the widely-adopted convention [1]}, [2]}, [3]}, [4]}, [5]} to cast the maximization problem into an expected-risk-based minimization problem \(f \in \operatornamewithlimits{argmin}_{f} R(f)\), where:
\(\begin{split}R(f) &= {\mathsf {MAUC}^\downarrow }= \sum _{i=1}^{N_C}\sum _{j \ne i}\frac{\operatornamewithlimits{\mathbb {E}}_{\mathbf {z}_1,\mathbf {z}_2}[\ell ^{i,j,1,2}_{0-1}|\mathcal {E}^{(ij)}]}{N_C(N_C-1)},\\\end{split}\) 
\(\begin{split}\ell ^{i,j,1,2}_{0-1} &= \ell _{0-1}(f^{(i)},\mathbf {x}_1,\mathbf {x}_2,y^{(i)}_{1},y^{(i)}_{2}) ,\\&= \mathbf {I}\left[ \Delta (y^{(i)}_{1,2}) \cdot \Delta (f^{(i)}) < 0 \right] + \frac{1}{2} \mathbf {I}\left[ \Delta (f^{(i)}) = 0\right] \end{split}\) 
